# O(ab) when o(a) and o(b) are relatively prime

• Chen
In summary: Then the order of that group is the order of the group of all infinite sequences whose elements are in that group. :smile:Thanks! In summary, the author is trying to prove that if two groups are relatively prime, and one is the order of a certain element in the other group, then the order of the first group is the order of the second group. He thinks he can do this by using the division theorem, but he is not sure if that is the right approach.
Chen
Hi,

I am trying to prove that if o(a) and o(b) are relatively prime, and ab = ba, then o(ab) = o(a)o(b). I'd appreciate it if someone could give me a nudge in the right direction because I've spent almost 2 days on this now and I got nowhere. Which is rather annoying considering this is the first exercice in the chapter and the rest I did without a problem, so there must be something simple here that I'm missing.

I already know that if (m, n) = 1 and m|k and n|k then mn|k. I think I can use this to prove what I need, if I can only show that o(a)|o(ab) and o(b)|o(ab). (Because I've already shown that o(ab)|o(a)o(b), so proving o(a)o(b)|o(ab) will be enough.)

Thanks!
Chen

Last edited:
What's o(x)? The order of x in the group of interest?

I think a better approach would be to show o(ab) >= o(a) o(b)

(P.S. I no longer think this approach is better!)

Last edited:
Yes. Any idea how I could go about showing that?

o(a) | o(ab) sounds like Lagrange's theroem!

As for o(ab) >= o(a) o(b), for some reason the division theorem springs to mind.

I thought Lagrange's theorem deals with orders of groups, not orders of elements...

It does. That's called a hint.

Sorry, I still don't get it. Which groups do you think should I define, to make use of Lagrange's theroem?

I tried defining H = <ab>, Ka = <a> and Kb = <b>. But then H is a subgroup of KaKb and the only thing I can learn from that is that the order of H (which is o(ab)) divides the order of KaKb (which is o(a)o(b)) - but I already know this...

And by the way, this chapter of questions comes before the chapter on Lagrange's theroem and even before the chapter on cyclic groups - so I think there's a way to solve this problem without making use of either of those.

Look at the group generated by ab, if a is in it, then you are done, or if yuou can show a^r is in it where r is some number coprime to o(a)...

suppose that r is a positive integer and a^rb^r=e, then a^r=b^s, for some positive s (ko(b)-r for some multiple of o(b)). If we raise both sides to the power o(b), then a^(ro(b))=e from which it follows that o(a) divides r as o(a) and o(b) are coprime, and hence o(ab), by symmetry o(b) divides o(ab) and we are done.

Thanks Matt, that did it. I took k=o(ab), then a^k=b^-k, raised to the power of o(b) etc.

By the way, while we're on the subject, can you guys think of any groups of infinite order, in which every element is of finite order? The only one I found was the group of all roots of unity in the complex with regular multiplication. Are there any more? (well I guess there are a lot more...)

Take your favorite finite group, and take the group of all infinite sequences whose elements are in that group. (pointwise multiplication)

## 1. What does "O(ab)" mean in this context?

In this context, "O(ab)" refers to the order of the group, which is the number of elements in the group. This notation is commonly used in group theory to denote the order of a group.

## 2. What does it mean for o(a) and o(b) to be relatively prime?

Two numbers are relatively prime if they do not have any common factors other than 1. In other words, their greatest common divisor is 1. In this context, it means that the orders of the two elements a and b do not share any common factors.

## 3. What is the significance of o(a) and o(b) being relatively prime?

The fact that o(a) and o(b) are relatively prime is significant because it implies that the group generated by a and b is a cyclic group, meaning that all the elements in the group can be written as powers of a and b. This makes it easier to understand and analyze the group's structure and properties.

## 4. How does the relative primality of o(a) and o(b) affect the order of the group O(ab)?

The order of the group O(ab) is equal to the product of the orders of a and b, i.e. o(a)*o(b). Therefore, if o(a) and o(b) are relatively prime, the order of the group O(ab) will be the product of two relatively prime numbers, which will be the same as their product. In other words, the order of the group will be the same as the order of the elements a and b.

## 5. Can o(a) and o(b) be relatively prime for any two elements in a group?

Yes, o(a) and o(b) can be relatively prime for any two elements in a group. However, this is not always the case. There are groups where the orders of all the elements are not relatively prime, meaning that they share common factors. In such cases, the group may not be cyclic and its structure may be more complex to analyze.

Replies
1
Views
774
Replies
25
Views
1K
Replies
3
Views
1K
Replies
5
Views
1K
Replies
4
Views
3K
Replies
2
Views
1K
Replies
3
Views
3K
Replies
1
Views
996
Replies
64
Views
4K
Replies
3
Views
9K